﻿ Haptic Exploration of Objects for Area Comparison and Measurement by Students with Visual Impairment: The Role of Objects’ Characteristics

# Haptic Exploration of Objects for Area Comparison and Measurement by Students with Visual Impairment: The Role of Objects’ Characteristics

By Chrysanthi Skoumpourdi and Maria Koza

Chrysanthi Skoumpourdi is a Professor in the Department of Sciences of Preschool Education and of Educational Design of the University of the Aegean in “Didactics of Mathematics: Designing Educational Material”. Her main research interest is related with teaching and learning mathematics to all the students (including students with visual impairment), with the use of educational materials, games and picture books.

Maria Koza is a primary school teacher and a PhD student in the Department of Sciences of Preschool Education and of Educational Design of the University of the Aegean. Her interest is focused on teaching mathematics to students with visual impairment.

## Abstract

Haptic exploratory movements of children with visual impairment are of educational and social interest because they can provide information as to how these children perceive the world around them in a tactile way, as well as how they construct knowledge. In this paper we analyzed the movements of primary school students with visual impairment, without accompanying problems, during their haptic exploration of objects, recognition of objects’ surface shape, and comparison and measurement of objects’ surface size. All data was video recorded. Every child, when individually met, carried out two activities. The results showed that the students recognized the shape of objects’ surfaces. Although they were able to compare the objects’ surface size when one object was longer and wider than the other, they encountered difficulty when each object differed in a different dimension (e.g., one was longer and the other one was wider). Moreover, students encountered difficulty during measuring the dimensions of the objects.

## Keywords

Visual impairment, haptic exploration, haptic measurement, surface, area

## Haptic Exploration of Objects for Area Comparison and Measurement by Students with Visual Impairment: The Role of Objects’ Characteristics

Haptic perception is the essential ability for understanding and interpreting the world by people with visual impairment. It combines primary and secondary sources of information. Touch and the movement and posture of the body are primary. Language issues, pre-existing knowledge, the kind of object/shape/activity, and the conditions under which the activity are realized are secondary (Argyropoulos, 2002).

The investigation of haptic movements through touch for children with visual impairment can reveal information as to how these children perceive the world around them and construct knowledge. Researchers’ investigations of the haptic movements of children focus on various issues that are of educational and social interest. Surface and its measurement are of educational interest because their understanding is the basis for various mathematical concepts and skills. They are of social interest because they are part of the culture of science, technology, and generally the everyday life of people (Bishop, 1991). A person uses the concept of surface and the size of surfaces in order to be able to communicate with peers, to do business, and improve his/her life (Struchens el al., 2003).

Participating in the dialogue of the research community on these issues, the present paper, which is a pilot case study, records and studies the movements of primary school children with visual impairment, without accompanying problems, during the haptic exploration of objects, the recognition of the shape of their largest flat surface, as well as the comparison and measurement of the size of their surface. In this study of the haptic exploratory movements of the students with visual impairment, particular importance was given to the design of the activities and the choice of the objects used, so that the influence of their characteristics on the movements of the students could be investigated. The following research questions were posed:

1. Which are the haptic movements used by students with visual impairment: (a) During the exploration of objects and recognition of the shape of their surface? (b) While comparing the size of the surface of objects? (c) While measuring the size of the surfaces of the objects?
2. Which characteristics of the objects influence the haptic movements of the students?

## Theoretical Framework: Haptic Perception

Haptic perception is a multi-sensory process related to the acquisition of information through touch to discover the composition of the object on the basis of its properties and characteristics (Lederman & Klatzky, 1987). The exploratory hand movements, that take place through the touch , determine what is perceived and the way it is perceived (Withagen et al., 2010). Together with movement and body posture, they constitute the primary sources of information for people with visual impairment. The secondary sources of information are spoken language issues, pre-existing knowledge, as well as the type and conditions under which an activity is realized. The primary and secondary sources of information, are interdependent and complement each other, forming a reference framework for the configuration of haptic perception for children with visual impairment (Millar, 1997), on which the spatial encoding, the perception of the surrounding space, as well as the exploration and recognition of objects, are based.

Two basic types of touch have been examined by researchers – passive and active (Gibson, 1962). During passive touch, the person obtains information without his initiative. In other words, it includes touching from other people or objects or unintentional touching himself (McLinden & McCall, 2002). During active touch, voluntary movements of the hands are carried out to explore and detect the surroundings or the object with the initiative, action, and participation of the person himself and is the most significant source of information for the perception of the environment and its peculiarities (Lederman & Klatzky, 1987; Roberts & Wing 2001; Withagen et al., 2010), especially for people with visual impairment.

## Techniques and Strategies of Students with Visual Impairment During Their Haptic Explorations

The most commonly used exploratory procedures are the following (Lederman & Klatzky, 1987): (a) enclosure, for a holistic approach to the objects, using both palms, (b) contour following, for a holistic approach while registering the detail of the outline of the objects, (c) lateral approach, for the texture of the objects, using the fingers when the object is small while using the palms when it is big, (d) pressure, to assess the hardness or softness of the objects, (e) static contact, to perceive the temperature of the objects, (f) unsupported holding, to perceive the weight of the objects, (g) function test, to realize the basic functions of the object and, (h) part motion test, for the specific function of the object.

To explore the objects and perceive their shape, students aged 11-13 used various haptic strategies such as: feeling the object mainly along its sides; rotating the object continuously with one or both hands; alternating the object from one hand to the other; clasping the object in the palm of the hand; holding the object from two of its corners with the left thumb and forefinger and the other two corners with the right thumb and forefinger; holding the object firmly on the desk with one hand (holding one corner of the object with the left hand, as the starting point) and counting the sides or the angles with the forefinger of the other hand; exploring the edges of the object with the tips of the fingers; pointing out the angles of the object with the forefinger; bringing the object very close to the face and/or touching the object with the cheek; making backward and forward movements between the face and the desk, with the object always in between (Argyropoulos, 2002).

The students use several haptic techniques and strategies to measure surface area (Mullet et al., 2012; Mullet & Miroux, 1996). If the children follow the height + width rule, then their haptic strategies include the measurement of these two dimensions. They place their thumb on one corner of the surface, and drag one finger of their hand until the end of this dimension. They implement the same technique with the second dimension of the rectangular surface. If the children implement the perimeter rule (Leon, 1982) to measure the surface, they place the finger of the one hand on a corner of the surface and they rotate the object clockwise or counterclockwise until they reach their starting point. If they follow the maximal extent rule, their strategy entails the measurement of the diagonal, or the bigger dimension of the surface, using one or both hands to measure the two most distant points. If they follow the area rule, they explore the surface with the tips of their fingers or their palms.

The number of fingers that participate in the identification of the object determine the field of view perceived by a child with visual impairment. Usually, the fingers used in the haptic process are the index finger, the middle finger, and the ring finger of both hands, while the thumbs and the little fingers are used mainly for support. Therefore, the haptic field of view is usually equal to six fingers, three from each hand (Berla & Butterfield, 1977).

## Measuring Area

Significant difficulties have been documented in the research, difficulties that originate from the curriculum, the school textbooks, the additional educational material used during the teaching procedure, and the way of teaching (Struchens et al., 2003). Research has shown that in activities that entail the measurement of the area, most of the mistakes originate from lack of understanding of the conceptual procedure (Barrantes & Blanco, 2006) and not from mistakes in the measurement itself (Huang & Witz, 2011). This is believed to occur due to the emphasis, given from the early years of education, on the use of and memorization of the area formula (Nunes et al., 1993).

## Method

### Participants

Four children born with visual impairment, without accompanying problems, took part in the study. They were attending two primary schools for blind children in Greece and knew the geometric shapes. Two students (A: female, B: female) were attending the fifth grade in Athens and were aged 11 and 12, respectively. The other two students (C: male, D: female) aged 12 and 10, respectively, were attending the fourth and second grades in Thessaloniki. Students A, B, and C had been taught the largest flat area formula of rectangular surfaces, while student D had not.

### Activities, Objects, and Educational Materials

The activities were designed and the objects were chosen bearing in mind the particularities of the children with visual impairment and on the results of earlier studies in the teaching of mathematics. In the first activity, an embossed picture (15 cm x 12 cm x 0.1 cm) and a picture frame without glass (20 cm x 17 cm x 0.5 cm) were presented and the following was asked: (a) “Can the picture fit in the frame?” and (b) “If we need to put glass in the frame, how much glass should we order?” In the second activity, two sesame candy bars were given, each having the same largest flat surface area but different perimeter (the first, 10 cm x 4 cm x 1 cm and the second  8 cm x 5 cm x 1 cm) and a question was asked: “Mr. George, the baker, makes these two types of sesame bars. Which one would you prefer and why?” In addition to the above, auxiliary materials were available that could be used by the children. These consisted of a square paper (10 cm x 10 cm) with 100 embossed squares, a paper strip (1 cm x 10 cm) with 10 embossed squares, a variety of square cardboards (1 cm x 1 cm, 2 cm x 2 cm, and 4 cm x 4 cm), as well as a pair of scissors. Also, an embossed set square1 and an embossed measuring tape were available for use, as the students were familiar with their use from classroom practice.

The design was to have all participants examine the same set of diverse objects. The features of the objects differed in respect to their size, thickness, weight, and whether they had an embossed surface or not (pattern or structure). The objects are as follows: (a) picture: thickness, very light (could move with a slight push), with embossed texture (pattern) on the surface and the size was about two palms; (b) picture frame: negligible thickness, not very light (could not move with a light push), without an embossed texture and the size of two palms; (c) sesame bars: thick, not very light (could not move with a light push), without an embossed texture, and the size was about one palm; and (d) auxiliary material: negligible thickness, very light, the small pieces were without an embossed texture, while the big pieces, up to the size of a palm, were with an embossed structure.

### Research Procedure

Every child participated individually in the pilot research and completed the two activities. Every task lasted about one teaching hour (approximately 45 minutes). All sessions were video recorded.2

## Results

Following are the results of the study, organized by student.

### Student A

To explore the picture, Student A held it steadily on the desk from the right side and scanned its surface using her left palm and fingers, except the thumb, to determine what it depicted. Then, using her index, middle, and ring finger of her left hand, she initially scanned along the left side and then the top and bottom sides. She used her thumbs to lift the picture from the desk and scan the sides using the fingers of her hands and palms. To ascertain the shape of the picture, she measured the length with the biggest opening of her thumb and middle finger and the width with the opening of her thumb and small finger of her left hand, realizing they were not equal. After that, she concluded that the picture was rectangular.

She explored the frame and its shape, initially by scanning its perimeter, and then by scanning its surface using both hands while the frame was steady on the desk. She estimated the length of the frame by using the biggest opening of the thumb and middle finger of her left hand. She estimated the width using the fingers of each hand in the center of the surface of the frame and pulling them horizontally towards the edges.

She initially answered that the picture could not fit into the frame, based on her previous exploratory movements. Thereafter, she placed the picture on the frame, trying to fit the left side of the picture and both of its sides with the corresponding left side and the corners inside the frame by simultaneously scanning them with the fingers of her left hand to ascertain that the picture could fit in the frame.

For the question of what glass size she should order, she measured the length of the frame with the biggest opening of her thumb and middle finger and the width of the frame with the opening of the thumb and small finger of her left hand. However, she could not give a numeric value for the size of the surface and she tried to use auxiliary material.

She explored the auxiliary material in different ways, depending on its size. She used both hands for the bigger pieces, holding them with her right and left hand from their right and left side, respectively, and scanned them simultaneously around the perimeter and internally with her palms and fingers. Her left hand was making more intense exploratory movements. She explored the smaller pieces by placing them in the palm of her left hand and scanned them by simultaneously rotating them with the tips of her fingers. The number of fingers involved was related to the size of the piece; the bigger the size the more fingers (thumb + index or middle or ring finger), the smaller the size the less fingers (thumb + index finger).

From the auxiliary material, she chose the 1 cm x 1 cm cardboard square, placed it on the bottom side of the frame and pushed it vertically from the bottom side to the top and vice versa, without coming to any conclusion. Later, she asked for the set square, measured the sides with the help of the researcher, and multiplied the numbers, thus finding the area of the surface inside the frame.

She explored the sesame bars placed on the desk. She grabbed the two sesame bars in both hands simultaneously, clasping them and moving her fingers up and down along the sesame bars. She alternated the simultaneous enclosure a few times scanning the surfaces of the sesame bars with the left hand and measuring their dimensions by opening her fingers as previously.

To compare the sesame bars, she did not place the one on top of the other at any moment. She measured the length of each sesame bar with the opening of the thumb and the index finger of each hand, respectively. With the index and small finger of both hands, she touched the right and left sides of the sesame bars at the same time to ascertain their width. Influenced by the longer length of the left sesame bar, she concluded that it was the biggest one. To verify her answer, she asked for the 4 cm x 4 cm cardboard square, placed it on the first sesame bar, and found it to be the same width (as it really was). Thereafter, she placed it on the second sesame bar and found it to be of the same width (the difference of one centimeter was lost). However, realizing that she could not ascertain their size with accuracy, she asked to use the set square. She measured the length and width of each sesame bar and implemented the rule of length x width, finding the area of the major surface. The girl would move her body back and forth when encountering difficulties during the whole procedure of the activities.

### Student B

To explore the picture and ascertain its shape, Student B held it steadily on the desk and scanned it with both hands simultaneously, using all the fingers except her thumbs, which were touching the bottom side, supporting or slightly lifting it. Thereafter, she checked its maximum extent. She touched the bottom left corner with the left hand and with the right hand, the upper right corner. To measure its width, she placed her thumb and middle finger of the left hand on the edges of the left side. She estimated the length by scanning the bottom side with the thumb of the right hand.
After that, she concluded that the picture was rectangular.

She explored the frame and its shape by scanning quickly with all the fingers of her left hand because she was holding the picture with her right hand.

In respect to the glass she had to order, she proposed to cut a cardboard of the same size. When she was asked to measure the area, she wanted to implement the area formula but she faced difficulty using the set square. With the help of the researcher, she measured and multiplied length and width.

To explore the auxiliary materials, she scanned them with her fingers. The size of the materials determined the number of fingers she used in combination with the thumb. She scanned with both hands when the researcher placed her hands on them.

She explored the sesame bars leaving her hands on them as the researcher had placed them. After grabbing them, she applied the opening created by the index finger and the thumb of her left hand at the corner of the first sesame bar and compared the size of its sides with the length of her fingers. With respect to the second sesame bar, she touched with her thumb and her index finger of her right hand the two diagonal tops, respectively, rotating it slightly and asserted that it was a square. To verify her answer, she enclosed the sesame bar with her hand, trying to estimate the width, having her thumb on one side and the rest of the fingers on the opposite side. She continued by touching the base of her palm on the bottom side and the index and middle finger on the opposite side while simultaneously touching the length of both sides with the thumb and small finger, respectively, and realized that the shape of the second sesame bar was rectangular.

She regarded the first sesame bar as “bigger” because it was longer, as she had ascertained from her former exploratory haptic moves. However, to justify her answer she used other haptic techniques. She placed the sesame bars lying on the desk next to each other but upright, aligning their bottom parts on the desk and re-ascertained the difference in length, but also the similarity in thickness. Thereafter, she placed the shorter sesame bar on top of the longer, but did not mention anything about its bigger width; probably the difference of one centimeter in width was lost due to their thickness. The researcher urged her to measure the sesame bars dimensions with a set square to be able to compare them, something the student did with the researcher’s help.

### Student C

Student C explored the picture by holding it with both hands, his thumbs in front and the rest of the fingers behind, and scanned it with the thumbs. Afterward, he placed it on the desk and scanned it simultaneously with all the fingers of both hands in the middle and all around in such a way that the picture was moving and rotating. Finally, he opened his palms and placed them on it, rotating it and then keeping it firm with his left hand on the upper left corner, scanning the outline with the index finger of his right hand, saying that the picture had a rectangular shape.

To explore the frame and its shape, he scanned it very quickly with all the fingers of both hands internally and around its perimeter. He held it momentarily with one hand, scanned with the other, and vice versa.

To ascertain if the picture can fit the frame, he initially placed his left palm in the position of the glass and moved it diagonally downward to understand the size of the space inside. Then he placed the frame upright on the desk and tried to position the picture, realizing that it did not fit. Afterward, with the encouragement of the researcher, he opened the rear part of the frame and placed the picture in it.

For the glass that he should order, he measured the length and the width of the space with his fingers and calculated their sum. Later, he measured the perimeter of the opening of the frame with four fingers joined (except the thumb) after measuring their length with the measuring tape. Every subsequent placement of the four joined fingers covered part of the previously measured part. In his next attempt, he placed both palms on the frame and said, “Ten fingers fit here,” “Two whole hands.”

To explore the auxiliary means, he scanned them simultaneously with all the fingers of both hands, but also holding firmly with one hand and scanning with the other. He counted the squares formed by the square decimeter using a string, starting the count of the squares having his hands on the right and left side of the square, from the outside to inside moving at the same time the index fingers of his hands until they met. He counted five when his fingers met, then doubled the number, and so he assessed that there were ten squares on each line.

From the auxiliary materials he chose the square decimeter, which he used as a measuring unit for the length and width of the frame and said, “I will tell the glazier to cut me a glass 23 wide and 14 long [meaning cm],” which was a correct answer.

He explored the sesame bars separately, alternating them from one hand to the other. With the thumb and index finger of his right hand, he checked the right angle and the length of the sides. Then, holding it in midair with both hands (with the big surface towards him), he joined his thumbs in the middle of the lower side and in the middle of the upper side he placed the index finger of his left hand and the middle finger of his right hand and dragged his fingers to the edges of the shape. He performed similar moves with the second sesame bar.

To compare the sesame bars, he initially placed them one next to the other and compared their length. He realized that they had the same thickness, but that one was longer. Afterward, he placed one sesame bar next to the other, joining the length of the second one with the width of the first one, and ascertained that the second sesame bar was “bigger.” Then he compared the length of the first one with the width of the second one. Next, he placed the one sesame bar on top of the other, but he could not come to a conclusion and asked for a measuring tape. He measured each one’s length and width with the help of the researcher; he added them up and found out that the first sesame bar was “bigger.” After being encouraged, the student remembered the area formula. He applied it and found that the two surfaces of the two sesame bars were equal. The movements of the student were very dynamic during all the activities and he alternated the different movements very quickly.

### Student D

Student D explored the picture by scanning initially its internal surface and then its outline. Holding it with her right hand and exploring its outline with the left, she assumed that the picture had a square shape. Trying to verify her answer, she held the bottom left-hand corner with her left hand and with her right thumb she touched the bottom right-hand corner, while simultaneously she opened the fingers of her right hand so that they ran across the length of the picture. By doing this she realized that the picture had a rectangular shape.

She explored the frame and its shape by holding it vertically with both hands and scanning it with her left hand until the top to ascertain its length. She explored the frame of the picture with her right hand, making side movements, shaking it from top to bottom, and hitting it in the middle of its surface. She clasped the top left corner of the frame in the palm of her left hand, while holding the bottom right hand corner with her right hand, then the two bottom corners, joining her hands in the middle of the bottom side and pulling them towards the edges, realized eventually that the picture was rectangular.

To ascertain whether the picture could fit in the frame, she placed the picture on the frame and realized that the picture could fit into it. To verify her answer, she explored the frame and the surface of the picture with her palms by making side movements. With her left hand she held the bottom left-hand corner of the picture and, touching her right thumb on the bottom right-hand corner, she stretched the rest of the fingers so that they could reach the other side in order to measure the length. Then, she stretched the fingers of the left hand to measure the width of the picture, while holding it steadily with the right hand. Realizing once more that it could fit in the frame, she opened it from the back and placed in the picture.

For the glass she should order, she pointed to the outline of the frame with her finger and said that “the glass should be so much.” When encouraged by the researcher to measure so that she could give an order to the glazier, she used the auxiliary material. She did not explore the auxiliary material; she simply touched it, ignored the small pieces, and placed the big pieces in the frame, moving them left and right. Thereafter, she asked for cardboard to cut it to the dimensions of the frame. Because the cardboard was big, she folded it and placed it in the frame, fit one corner and one side in the respective corner and side of the frame, and gave it to the researcher to cut it.

To explore the sesame bars, the student grabbed them with both hands at the same time. She touched the first sesame bar with her palm and distinguished its sides by scanning it with her index finger from start to end, recognizing its rectangular shape. She twirled the second sesame bar in her hand, clasped it in both hands, and recognized it as rectangular. However, when she compared the second sesame bar against the first one, she changed her opinion and regarded it as having a square shape. Trying to verify her thinking, she held the upper left corner with the thumb and the index finger of her left hand and with the index finger of her right hand she traced its width and length. Then, she traced with the thumb of her left hand across the left side, from the top left corner to the bottom left corner. Finally, she held the bottom right and left corners of the sesame bar with her thumbs and the upper right and left corners with her index fingers; she ascertained that it was a rectangle.

To compare the sesame bars, she placed them on the desk next to each other, aligning their bottom sides. With the palm of her right hand, she touched the two top sides, saying that the “bigger” one was the one which was longer. To justify her answer, she held the two bars vertically next to each other with the bottom sides firmly touching the desk while their sides were joined. Next, she grabbed the second sesame bar with both hands, as if she was assessing its thickness, and placed it over the first one, tracing its length with the thumb of her left hand. She held the two sides of the sesame bars firmly in one hand, which was vertical on the desk, while they were next to each other and compared their backs. This time, she regarded the second sesame bar “bigger” because it was wider. And, although she discovered that the first was longer and the second one was wider, she concluded that the first one was the “bigger” sesame bar.

The movements of the student during the activities were erratic; she was tapping her fingers regularly on the objects, moving them left and right.

## Discussion and Conclusions

The movements of the students during the haptic exploration of the objects, the recognition of their surface shape, as well as the comparison and measurement of their largest flat surface size were made mainly with the fingers and the palms, with the object at times lying on the student’s desk and, at times, in midair. Although some of these movements are included in the ones described by Lederman and Klatzky (1987), by Mullet et al. (2012), as well as by Argyropoulos (2002), there is not any research data about the specialization of the movements consistent with the type of activity, the aim of the activity, and the different characteristics of the objects – something that was detected from the current research data.

The fingers used during the exploration of the objects and the area measurement were, in general, the index, the middle, and the ring finger; the same as mentioned by Berla and Butterfield (1977). But, in some instances, the children used the rest of their fingers and their palms, while they used their thumbs not only to support, but also to lift the objects. The number of hands involved, the fingers used, and the placing and the manner of scanning appeared generally suited to the features of the objects.

To explore the objects that had an embossed texture on their surface, they first scanned the surface and then scanned the outline of the object. The movements they used to explore the picture and the auxiliary materials with negligible weight and thickness, with the size of two and one palms, respectively, and an embossed texture were the following (the children who used them are mentioned in the parentheses):

1. Holding the object firmly on the desk with the right hand and scanning with the left, using the palm and mainly three fingers (the index, middle, and ring fingers). Slightly lifting the object from the bottom side and scanning the sides with the rest of the fingers and the palms (A and D).
2. Scanning with both hands simultaneously using all the fingers, except the thumbs. The thumbs were touching the bottom part of the object and sometimes supporting or lifting the object slightly (B).
3. Scanning with the thumbs, holding the object in both hands and then with all the fingers of both hands, while the object was lying flat on the student’s desk (C).

The opposite happened with objects without an embossed texture. The children first scanned the perimeter of the object and then (or simultaneously) scanned the surface. Moreover, their moves were different depending on the object. The frame, which was not a very light object with negligible thickness and as big as two palms, was explored in the following way:

1. Scanning with the thumb, index, and middle finger of both hands, with more intense movements of the left hand, while the frame was lying flat on the desk (A).
2. Scanning with all the fingers of the left hand while it was lying flat on the desk (B).
3. Simultaneously scanning the perimeter and interior with all the fingers of both hands very quickly. Momentarily holding it with one hand and scanning with the other and vice versa (C).
4. Scanning with both hands, first with the left and then with the right, making side movements and tapping its surface, holding it vertically with both hands (D).

The auxiliary materials, which were very light, with negligible thickness and small size (1 cm x 1 cm, 2 cm x 2 cm, and 4 cm x 4 cm), were explored by:

1. Scanning with two (thumb + index), with three (+ middle), or four (+ ring) fingers of the left hand, depending on their size (A).
2. Scanning with two (thumb + index), three (+ middle), or four (+ ring) fingers of both hands, depending on their size (B and C).

The sesame bars, which were not very light objects, with a thickness of 1 cm, and largest flat area surface of about one palm, were explored by all the children while the bars were lying on the desk. They clasped the bars in their hands and scanned them across, up, and down with one or more fingers, as well as with their palm. They gripped them within their palm and fingers, but also by placing the thumb on the one side and the rest of the fingers on the other side. The gripping was alternated a few times with scanning of the surface of the sesame bars.

To recognize the shape of the surfaces of the objects, the children estimated comparatively the dimensions of two continuous sides by either non-simultaneous explorations of the sides of the object – the length of the object was examined and then the width – or simultaneous exploration of the sides of the object. For the former, they measured the length of all the objects by opening their fingers (thumb + index, or thumb + middle) of the left hand (A, B) or the right hand (D).

The estimation of the width of small objects with thickness (sesame bars) was carried out by holding the right and left side of the bar with the index and the small finger of the left hand (A).

The measurement of the width of the picture, as a big lightweight object, was done by:

1. Opening of the fingers (thumb + small) of the left hand (A).
2. Scanning the bottom side with the thumb of the right hand (B).

The measurement of the width of the frame, as a big lightweight object, was done by:

1. Joining the fingers of each hand in the middle of the surface and horizontally pulling towards its edges (A).
2. Joining the thumbs in the middle of the bottom side and placing the index finger of the left hand and the middle finger of the right hand in center upper part of the frame and pulling to the edges (C).
3. Joining the fingers of each hand in the center of the bottom side and pulling horizontally to its edges (D).

Simultaneous measurement of length and width was made for the picture by:

1. Placing the left palm on the bottom left-hand corner and the right palm on the upper right corner (A), or placing the left palm on the upper left corner and the right palm on the bottom right corner (D), known as the rule of the longest extent.
2. Placing the palms on the surface (C), known as the rule of the surface.
3. Stabilizing the object with the left hand on the upper left corner and scanning its outline with the index finger of the right hand (C), known as the rule of the perimeter.
4. Scanning the upper right side with the index finger of the right hand and the left side with the thumb of the left hand (D).

Also, a simultaneous measurement of length and width was effected with the sesame bars, regarded as small objects, by:

1. Matching the corner and the size of the object’s sides with the corner and the size of the thumb and the index finger of the left (B) or right hand (C).
2. Placing the thumb and index finger of the right hand on the two diagonal sides respectively (B).
3. Placing the bottom part of the palm on the bottom side, the index and middle finger on the opposite side, and the thumb and little finger on the left and right sides, respectively (B).

As far as the comparison of the objects with the size of two palms and with the one object being longer and wider from the other (i.e., the frame, the picture), the children usually placed the smaller object on the bigger one, matching them on the one side and a corner. However, the comparison of objects with the size of about a palm, which differed in both dimensions, and with the one being longer (2 cm taller) and the other wider (1 cm wider) made it hard for the students. All of them placed the wider object on the longer one and concluded that the longer sesame bar was the “bigger” one and that was the one they would choose.

But many other comparisons came about in their attempt to justify their answer:

1. Comparing the length and thickness of the objects by placing them next to each other while lying on the desk and aligning them at the bottom part to ascertain the length (A, B, C, and D).
2. Comparing the length and the thickness of the objects by placing them next to each other, as far as the length is concerned, upright and aligning the bottom part on the desk (B).
3. Comparing the length and the thickness of the objects by placing them next to each other, as far as their surface is concerned, upright and aligning the bottom part on the desk (D).
4. Comparing the length and width of the objects by placing the shorter on top of the longer one (B and C).
5. Comparing the width and thickness of the objects by placing them next to each other, as far as the width is concerned, lying on the desk and aligning their bottom part (C).
6. Comparing the width to length and thickness of the objects by placing them next to each other in respect to the width and in respect to their length by aligning their bottom sides on the desk (C).
7. Comparing the length to width and thickness of the objects by placing them next to each other, the one as far as its length is concerned and the other as far as its width, by placing them on the desk and aligning the bottom sides (C).
8. Comparing with the use of mediating auxiliary material (A).

The variety of the above strategies probably arose from the difficulty of the comparison of the largest flat surface area. Student D, the only one who had not been taught the area formula, found out that the one sesame bar was longer while the other one was wider, but could not continue her thinking and assumed that the longer was also bigger. The same answer was given by the rest of the children. Student C measured the surface of each sesame bar separately, using the mathematical formula, to discover that they both have the same largest flat surface area.

Measuring the area of the objects was very hard for the children, even for those who had been taught the area formula. They tried to estimate the size of the area in unconventional ways (i.e., measuring with auxiliary materials, approximate measurement with their fingers, measuring by comparing using intermediary material) as much as in conventional strategies (i.e., measuring with a set square or measuring tape). None of the children partitioned the area to measure its size, nor used the auxiliary material effectively. The limitations in the students’ understanding of the conceptual process to find the area were obvious and this is also mentioned by Huang and Witz (2011). The answers given for the dimensions of the glass were:

1. By means of intermediary material – cardboard cut to the dimensions of the frame (D).
2. By means of a numeric value which was the result of an approximate measurement, the sum of the two dimensions (the rule of length + width), with the use of the fingers (C).
3. By means of a numeric value which corresponded to the area, which was the result of measurements of length and width of the objects with conventional tools and with the help of the researcher (A, B, C).

In conclusion, in this paper in which the movements of children with visual impairment were recorded and analyzed during their haptic exploration of objects in order to perceive the shape of their surface, the comparison, and the measurement of the size of their largest flat surface area, it was clear that the children realized with ease which of the object’s flat surfaces is largest and that it is rectangular. Even in cases where they mistakenly believed the rectangle was a square, they had the techniques and strategies to verify or reject their claim. Moreover, they compared the object’s surface size of about two palms and negligible third dimension, one of which was longer and wider than the other. However, they encountered great difficulty when comparing the objects’ surfaces, of about one palm size with a non-negligible third dimension, which differed for one in length and the other in width, as well as when measuring the area of the objects. Regarding the role of objects’ characteristics, it seemed that they influenced the children's movements, which differed in terms of the hands and the number of fingers that were used, as well as the way the scanning took place. Both hands were involved with objects of the size of two palms, while only one hand was involved with smaller objects. With respect to objects with thickness, the children placed their fingers on the surface of the thick part of the object or gripped it while lying on the desk. Also, the existence of this dimension made it difficult for the children to estimate their size by directly comparing their largest flat surfaces. Objects that moved with a light push because of their insignificant weight were supported with the one or both hands and scanned with the other hand or with the rest of the fingers. Objects with an embossed texture, pattern, or structure on their surface motivated the children to start exploring the object from the pattern or structure.

## Implications for Practitioners and Families

The recording and analyzing of the different haptic movements of the children with visual impairment during the realization of mathematical activities in respect to the target and type of the activity and the material, lead to the understanding of the haptic behavior of each child and to the discovery of the variety of their haptic approaches. This data is useful for the educational and research design of educators and researchers, respectively, as well as for families. Knowing which movements are effective for haptic explorations of objects, for object’s shape recognition, and for measurement of the area of the objects could be included by the educators in their teaching plan, as well as by the families to their daily practice, so as to encourage, reinforce, and support the explorations and the measurements of children with visual impairment. Additionally, knowing how specific objects’ characteristics influence children’s movements could inform educational communities and families in order to invent and cultivate different exploration techniques and strategies with their children. Finally, in knowing the difficulties that children with visual impairment face when dealing with haptic exploration of objects in order to perceive the shape of their surface, the comparison, and the measurement of the size of their largest flat surface area could inform educational communities and families in order to design the appropriate activities to overcome these difficulties.

## Future Research

The sample of the students was small; therefore, it is deemed necessary to enrich the results of the specific research with techniques of other children, so as to map the way by which the students with visual impairment recognize, compare, and measure the surfaces through haptic movements. More research is needed to investigate the techniques and strategies children with visual impairment use during the haptic exploration of objects, the recognition of the shape of their largest flat surface, as well as the comparison and measurement of the size of those surfaces in which the activities and the objects would have different characteristics from the above. Furthermore, aspects that are not covered in this particular research and thus limit the generalization of its results, such as the usage of other types of activities and objects, other shapes, other materials, different texture (e.g., abrasive, glossy, matte, embossed, velvet, soft, hard), and other dimensions of objects, ranging from those that fit in the palm of a hand to the floor of an actual room, could give additional data about the way the children with visual impairment explore, recognize, compare, and measure largest flat surface areas. Moreover, further associations could be made. For instance, which techniques are favored in the recognition, comparison, and measurement of the surfaces, taking into account whether they are even or abrasive, etc., when the surfaces are bigger than their body, when the surfaces are heavy or light, etc. In addition, the results of these mapping techniques could be compared with results coming from children of the general population.

## References

Argyropoulos, V. (2002). Tactual shape perception in relation to the understanding of geometrical concepts by blind students. British Journal of Visual Impairment, 20(1), 7-16. https://doi.org/10.1177/026461960202000103

Barrantes, M., & Blanco, L.J. (2006). A study of prospective primary teachers’ conceptions of teaching and learning school geometry. Journal of Mathematics Teacher Education, 9(5), 411-436. https://doi.org/10.1007/PL00021938

Berla, E.P., & Butterfield, L.H. (1977). Tactual distinctive features analysis: Training blind students in shape recognition and in locating shapes on a map. Journal of Special Education, 11(3), 335-346. https://doi.org/10.1177/002246697701100309

Bishop, A. (1991). Mathematical enculturation. A cultural perspective on mathematics education. Kluwer Academic Publishers.

Gibson, J.J. (1962). Observations on active touch. Psychological Review, 69(6), 477-491. https://doi.org/10.1037/h0046962

Huang, H.-M.E., & Witz, K.G. (2011). Developing children’s conceptual understanding of area measurement: A curriculum and teaching experiment. Learning and Instruction, 21(1), 1-13. https://doi.org/10.1016/j.learninstruc.2009.09.002

Lederman, S.J., & Klatzky, R.I. (1987). Hand movements: A window into haptic object recognition. Cognitive Psychology, 19(3), 342-368. https://doi.org/10.1016/0010-0285(87)90008-9

Leon, M. (1982). Extent, multiplying, and proportionality rules in children's judgments of area. Journal of Experimental Child Psychology, 33(1), 124-141. https://doi.org/10.1016/0022-0965(82)90010-8

McLinden, M., & McCall, S. (2002). Learning through touch: supporting children with visual impairments and additional difficulties. David Fulton Publishers.

Mullet, E., Martinez, G.-E.-M., Makris, I., Rogéd, B., & Sastred, M.-T.-M. (2012). Functional measurement: An incredibly flexible tool. Psicológica, 33(3), 631-654. https://files.eric.ed.gov/fulltext/EJ980498.pdf

Mullet, E., & Miroux, R. (1996). Judgment of rectangular areas in children blind from birth. Cognitive Development, 11(1), 123-139. https://eric.ed.gov/?id=EJ523497

Nunes, T., Light, P., & Mason, J. (1993). Tools for thought: The measurement of length and area. Learning and Instruction, 3(1), 39-54. https://doi.org/10.1016/S0959-4752(09)80004-2

Roberts, R., & Wing, A.M. (2001). Making sense of active touch. British Journal of Visual Impairment, 19(2), 48-55. https://doi.org/10.1177/026461960101900202

Struchens M.E., Martin W.G., & Kenney P.A. (2003). What students know about measurement: Perspectives from the NAEP.In D. H. Clements & G. Bright (Eds.), Learning and teaching measurement (pp. 197-208). Reston: NCTM.

Withagen, A., Vervloed, M.P.J., Janssen, N.M., Knoors, H., & Verhoeven, L. (2010). Tactile functioning in children who are blind: A clinical perspective. Journal of Visual Impairment and Blindness, 104(1), 43-54. https://doi.org/10.1177/0145482X1010400108

1. A right-angled triangular plate for drawing lines, especially at 90°, 45°, 60°, or 30°

2. Procedures of our local Institutional Review Board to ensure the safety of research subjects have been followed. A special permission was granted by the relevant authorities, the students themselves, their parents/legal guardians, as well as the teachers for video recording. Moreover, the research was designed according to the ethics for research with blind students. Video recordings are accessed only for author’s PhD research.

The Journal of Blindness Innovation and Research is copyright (c) 2021 to the National Federation of the Blind.